Model Calculation Seminars

Quarterly seminars open to anyone where we talk about any subject related to scientific computation in condensed matter physics.

Please register at least one day before the seminar. Once registered, you will receive information on upcoming seminars.

The 9th MCSeminar

Exact Renormalization Flow for Matrix Product Density Operators

Speaker: Kohtaro Kato (Graduate School of Informatics, Nagoya University)
Time & Date: 15:30 – 16:30, September 30 (Monday), 2024 (JST)
Place: Hybrid (A615 / Zoom meeting)

Matrix Product Density Operators (MPDOs) provide an efficient tensor-network representation of mixed states in one-dimensional quantum many-body systems, often used to describe thermal states and steady states in dissipative dynamics. While MPDOs generalize Matrix Product States (MPS), which effectively describe 1D pure states and are associated with gapped ground states, the renormalization properties of MPDOs are more complex.
In this work, we investigate real-space renormalization group (RG) transformations of MPDOs, modeled as circuits of local quantum channels. We impose that the renormalization flow must exactly preserve correlations between coarse-grained sites, ensuring that it is invertible through other local quantum channels. We first show that unlike MPS, which always admit well-defined isometric renormalization flows, MPDOs generally do not exhibit such exact flows. We introduce a subclass of MPDOs with well-defined renormalization flows, showing that these states possess a coalgebra structure and obey generalized symmetries described by Matrix Product Operator (MPO) algebras. Additionally, we explore the fixed points of these renormalization flows, providing insights into the classification of mixed-state quantum phases and the role of non-invertible symmetries in this subclass of MPDOs.

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The 8th MCSeminar

Probability control nonreversible Markov chain Monte Carlo

Speaker: Hidemaro Suwa (Department of Physics, The University of Tokyo)
Time & Date: 13:00 – 14:00, April 11 (Thr), 2024 (JST)
Place: Hybrid (A614 / Zoom meeting)

In recent developments, Monte Carlo methods that strategically break detailed balance to manipulate the flow of probabilities have emerged [1]. These methods include optimizing the transition probabilities during state updates. We developed an optimization algorithm designed to minimize the rejection probability and successfully applied it to various statistical mechanical models, such as the Potts model and quantum spin systems [2]. Further, we introduced an algorithm capable of controlling the rejection rate through a single parameter, revealing that reducing the rejection rate leads to an exponential increase in computational efficiency [3]. Another intriguing strategy for breaking detailed balance is the concept of lifting, which expands the state space to introduce probability flow in the enlarged state space. The lifting technique is particularly effective in particle systems, as demonstrated by the event chain Monte Carlo method [4]. In this talk, reviewing these approaches to constructing nonreversible Markov chains, we will present the lifted directed-worm algorithm [5,6] and the multi-replica swap optimization of the replica exchange method (namely, parallel tempering). These probability control nonreversible Markov chains significantly improve the computational efficiency of Monte Carlo sampling.

[1] H. Suwa and S. Todo, Butsuri 77(11) 731-739 (2022).
[2] H. Suwa and S. Todo, Phys. Rev. Lett. 105, 120603 (2010).
[3] H. Suwa, Physica A 633, 129368 (2024).
[4] W. Krauth, Front. Phys. 9:663457 (2021).
[5] H. Suwa, Phys. Rev. E 103, 013308 (2021).
[6] H. Suwa, Phys. Rev. E 106, 055306 (2022).

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The 7th MCSeminar

The Self-Learning Monte Carlo Method: Accelerating Simulations with Machine Learning

Speaker: Yuki Nagai (Japan Atomic Energy Agency)
Time & Date: 13:00 – 14:00, December 26, 2023 (JST)
Place: Hybrid (A614 / Zoom meeting)

The Self-Learning Monte Carlo Method: Accelerating Simulations with Machine Learning

We have introduced a general method, dubbed self-learning Monte Carlo (SLMC), which speeds up the MC simulation by designing and training a model to propose efficient global updates. We have developed the SLMC in various kinds of systems for electrons[1], spins[2], atoms[3], and quarks and gluons[4].

For example, we proposed an efficient approach called self-learning hybrid Monte Carlo (SLHMC) method, which is a general method to make use of machine learning (ML) potentials to accelerate the statistical sampling of first principles density-functional-theory (DFT) simulations[3]. In the SLHMC simulation, the statistical ensemble is sampled exactly at the DFT level for a given thermodynamic condition. Meanwhile, the ML potential is improved on the fly by training to enhance the sampling, whereby the training dataset, which is sampled from the exact ensemble, is created automatically.

In this talk, I will show the basic concept of SLMC and various kinds of applications.

[1] YN, H. Shen, Y. Qi, J. Liu, and L. Fu, Self-Learning Monte Carlo Method: Continuous-Time Algorithm, Phys. Rev. B 96, 161102 (2017).; YN, M. Okumura, K. Kobayashi, and M. Shiga, Self-Learning Hybrid Monte Carlo: A First-Principles Approach, Phys. Rev. B 102, 041124 (2020).
[2] H. Kohshiro and YN, Effective Ruderman–Kittel–Kasuya–Yosida-like Interaction in Diluted Double-Exchange Model: Self-Learning Monte Carlo Approach, J. Phys. Soc. Jpn. 90, 034711 (2021).;YN and A. Tomiya, Self-Learning Monte Carlo with Equivariant Transformer, http://arxiv.org/abs/2306.11527.
[3] YN, M. Okumura, K. Kobayashi, and M. Shiga, Self-Learning Hybrid Monte Carlo: A First-Principles Approach, Phys. Rev. B 102, 041124 (2020).;K. Kobayashi, YN, M. Itakura, and M. Shiga, Self-Learning Hybrid Monte Carlo Method for Isothermal-Isobaric Ensemble: Application to Liquid Silica, J. Chem. Phys. 155, 034106 (2021).
[4] YN, A. Tanaka, and A. Tomiya, Self-Learning Monte Carlo for Non-Abelian Gauge Theory with Dynamical Fermions, Phys. Rev. D (2023).;Y. Nagai and A. Tomiya, Gauge Covariant Neural Network for 4 Dimensional Non-Abelian Gauge Theory, http://arxiv.org/abs/2103.11965.

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Spin-lattice glass transition without quenched disorder on pyrochlore magnet

Speaker: Kota Mitsumoto (IIS, Univ. Tokyo)
Time & Date: 13:00 – 14:00, September 22, 2023 (JST)
Place: Hybrid (A614 / Zoom meeting)

In the community of magnetism, it has been generally assumed that spin glass transitions occur due to frustration from random interactions, so-called quenched disorder. However, disorder-free spin glass transitions are observed experimentally in a geometrically frustrated pyrochlore magnet. So far, there are no convincing theoretical explanations for the mechanism of this spin glass transition without quenched disorder.

A recent experiment suggested that lattice distortions play important roles in the spin glass transition on the prototypical geometrically frustrated spin glass Y2Mo2O7[1]. This lattice distortion results from the selection of the electron orbitals[2], i.e., Jahn-Teller distortion. Being motivated by the experiment, we introduced a model which includes not only the spin degrees of freedom but also the lattice distortions as dynamical variables. This model doesn’t include any quenched disorder, but both spins and lattice distortions are geometrically frustrated. We performed extensive numerical simulations for the model and analyzed a mean-field model which can be solved exactly in the infinite dimension.

In the numerical simulations[3], we found that spins and lattice distortions simultaneously freeze at a common finite temperature. Both degrees of freedom do not exhibit any long range order below the freezing temperature. In the mean-field analysis in the spherical limit using the replica method[4], we found that replica symmetry breaking appears only in the phase where both spins and lattice distortions are frozen, implying that a complex free-energy landscape is induced by the spin-lattice coupling

References
[1]P. M. Thygesen, et al., Phys. Rev. Lett. 118, 067201 (2017)
[2]KM, C. Hotta and H. Yoshino, Phys. Rev. Research 4, 033157 (2022)
[3]KM, C. Hotta and H. Yoshino, Phys. Rev. Lett. 124, 087201 (2020)
[4]KM and H. Yoshino, Phys. Rev. B 107, 054412 (2023)

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Zeros of Green functions in topological insulators – A tool for visualizing topological phases

Speaker: Takahiro Misawa (ISSP)
Time & Date: 13:00 – 14:00, May 15, 2023 (JST)
Place: Online (Zoom meeting)

Recently, we have found that the zeros of the diagonal components of the Green functions are useful quantities for detecting a wide range of topological insulators [1]. In particular, we have shown that the zeros of the Green functions traverse the band gap due to band inversions in the topological phases. Utilizing this feature, we can distinguish topological phases by seeing whether the zeros traverse the band gap. For microscopic models of the conventional six classes of topological insulators, we show that the traverses of the zeros universally occur in the topological phases. We also show that higher-order topological insulators, which have recently attracted much attention, can also be detected by the zeros of the Green functions.

Interestingly, the recently rediscovered eigenvector-eigenvalue identity [2], which is a simple but long-time-overlooked mathematical formula in linear algebra, plays an important role in the analysis of the zeros of the Green functions. Furthermore, by using the zeros of the Green function, we find that a conventional antiferromagnetic Mott insulator in κ-(BEDT-TTF)2Cu[N(CN)2]Cl can be regarded as a correlated topological insulator [3].

References
[1] T. Misawa and Y. Yamaji, Phys. Rev. Research 4, 023177 (2022).
[2] P. Denton, S. Parke, T. Tao, and X. Zhang, Bull. Am. Math. Soc. 59, 31 (2022). The story on the finding of the eigenvector-eigenvalue identity is available at https://www.quantamagazine.org/neutrinos-lead-to-unexpected-discovery-in-basic-math-20191113/
[3] T. Misawa and M. Naka, arXiv:2301.04490.

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Monte Carlo sampling in tensor-network representation

Speaker: Synge Todo (University of Tokyo)
Time & Date: 13:00 – 14:00, February 20, 2023 (JST)
Place: Online (Zoom meeting)

Many classical and quantum lattice models can be represented as tensor networks. However, the exact contraction of a tensor network is generally exponentially expensive, and some approximation is usually required. In numerical simulations based on the tensor networks, approximations with the singular value decomposition are widely used. On the other hand, various contraction methods based on randomized algorithms have also been proposed. Unfortunately, with a simple weighted sampling, it is difficult to control the accuracy because the expected value variance diverges rapidly as the network grows. In this talk, I propose a new tensor contraction method based on Monte Carlo sampling. The proposed method combines the stochastic basis transformation of tensors with the Markov chain Monte Carlo framework. It can entirely remove the systematic error due to a finite bond dimension in the approximate tensor-network contraction while controlling the variance of measurements.

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Numerical analysis of the Sachdev-Ye-Kitaev type models

Speaker: Masaki Tezuka (Kyoto University)
Time & Date: 13:00 – 14:00, November 21, 2022 (JST)
Place: Online (Zoom meeting)

The Sachdev-Ye-Kitaev (SYK) model, proposed in 2015, is a quantum mechanical model of N Majorana or complex fermions with all-to-all random interactions. The model has attracted significant attention over the years due to its features such as the existence of the large-N solution with maximally chaotic behavior at low temperatures and holographic correspondence to a low-dimensional gravity theory. Numerical diagonalization of the model Hamiltonian gives much insight into the dynamics of the model. We have studied (i) the spectral correlation by computing the spectral form factor [1], and (ii) the scrambling dynamics of the model by using the Hayden-Preskill protocol [2]. In addition to the result for the original SYK model, we present the results for the binary-coupling sparse SYK model [3] and the SYK4+2 model [4].

References
[1] J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S. H. Shenker, D. Stanford, A. Streicher, and M. Tezuka, JHEP 1705:118 (2017) [arXiv:1611.04650]; H. Gharibyan, M. Hanada, S. H. Shenker, and M. Tezuka, JHEP 1807:124 (2018) [arXiv:1803.08050].
[2] P. Hayden and J. Preskill, JHEP 0709:120 (2007) [arXiv:0708.4025].
[3] M. Tezuka, O. Oktay, E. Rinaldi, M. Hanada, and E. Nori, arXiv:2208.12098.
[4] A. M. García-García, B. Loureiro, A. Romero-Bermúdez, and M. Tezuka, Phys. Rev. Lett. 120, 241603 (2018) [arXiv:1707.02197].

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Variational Tensor Network Operator

Speaker: Yu-Hsueh Chen (National Taiwan University)
Time & Date: 13:00 – 14:00, August 1, 2022 (JST)
Place: Online (Zoom meeting)

We propose a simple and generic construction of the variational tensor network operators to study the quantum spin systems by the synergy of ideas from the imaginary-time evolution and variational optimization of trial wave functions. By applying these operators to simple initial states, accurate variational ground state wave functions with extremely few parameters can be obtained. Furthermore, the framework can be applied to study spontaneously symmetry breaking, symmetry protected topological, and intrinsic topologically ordered phases, and we show that symmetries of the local tensors associated with these phases can emerge directly after the optimization without any gauge fixing. This provides a universal way to identify quantum phase transitions without prior knowledge of the system.

Reference
Y.-H. Chen, K. Hsu, W.-L. Tu, H.-Y. Lee, and Y.-J. Kao, arXiv:2207.01819

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Probing the Vicinity of Deconfined Quantum Critical Point with Quantum Monte Carlo

Speaker: Jun Takahashi (U. New Mexico)
Time & Date: 10:00 – 11:00, April 28, 2022 (JST)
Place: Online (Zoom meeting)

Deconfined quantum criticality (DQC) is a beyond-Landau paradigm quantum phase transition predicted to occur in a 2d quantum magnet between a Neel phase and a valence bond solid (VBS) phase. Although the two phases are not related in the traditional symmetry-breaking perspective, field theory predicts that at the DQC transition point the order of the phases fractionalizes and creates an emergent gauge field, letting the Neel-VBS transition generically continuous.

By constructing concrete models that could be simulated to a large scale with quantum Monte Carlo methods, we numerically study the Neel-VBS transition and find that there is an emergent higher symmetry at the first-order transition point up to a very large system size. We also find that there are a number of new perturbations to the DQC point that were previously overlooked, and show that they lead to a different phase diagram, possibly resolving the previously observed discrepancy between theoretical conformal bootstrap calculation and numerical studies.

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